The telephone polynomials: An Appell-type orthogonal polynomials connecting Hermite–Laguerre polynomials

This article investigates a new Appell-type sequence, the telephone polynomials, which extend the classical telephone (involution) numbers. We present their fundamental algebraic properties, structural characterizations, and diverse interconnections with well-known families such as the probabilists’ and physicists’ Hermite polynomials, generalized Laguerre polynomials, and Legendre polynomials. Through the established telephone-Hermite relationships, several new identities involving Laguerre and Legendre polynomials are derived. Furthermore, we demonstrate that the telephone polynomials possess the Appell sequence property, satisfy an additive identity over variables, and obey a specific differential equation and integral representation. Notably, these polynomials are shown to be orthogonal with respect to the weight function ex2/2 along the imaginary axis, an orthogonality property analogous to that of the Hermite polynomials. Additional orthogonal-type properties are also explored and discussed in detail.